Linear Algebra
Matrix Multiplication
Multiplying two matrices chains their maps: AB means apply B, then A. Run both orders side by side and watch the letter F end up in two different places.
ABimage of îimage of ĵthe F
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Order matters?—
—
A
·B
=AB
Matrix A
1.001.000.001.00
Matrix B
0.00-1.001.000.00
Show
noneboth
Values
det A—
det B—
det AB—
det BA—
AB—
BA—
What to observe
- Take A = Shear and B = Rotate 90° and run both orders. Rotating and then shearing leaves the F somewhere else than shearing and then rotating, so AB ≠ BA.
- Two rotations commute: 90° then 45° is 45° then 90°, and both sides end on the same F. Two diagonal stretches commute too, since each just scales the axes.
- Set B = A⁻¹. B undoes the shear before A redoes it, so the F comes back home and AB is the identity.
- Watch the area factor in each panel: B scales areas by det B, then A by det A, so det AB = det A · det B in either order. A reflection makes it negative and flips the F.
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